Nonlinear dynamics of divide-by-two injection-locked frequency dividers in locked operation mode

摘要

A method for analyzing the nonlinear dynamics of the injection-locked frequency dividers in synchronized operation mode is presented, including the stability analysis of locked states. We use a specific divide-by-two circuit, namely a differential LC CMOS divider with a complementary topology, as a guideline for presentation, showing that the sizing of the devices significantly affects the synchronization mechanism of the divider, which exhibits a very rich dynamical behavior. We provide closed-form expressions to determine the amplitude and the phase in the locked state, as well as the locking range, leading to accurate results, which are validated by numerical simulations. The presented analysis of the frequency divider dynamics enables us to establish that stable locked oscillations occur on the whole locking range predicted by the well-known Adler's equation and that these are possible also beyond that range.